231 lines
12 KiB
HTML
231 lines
12 KiB
HTML
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<html lang="zh" dir="ltr" class="client-nojs" xmlns="http://www.w3.org/1999/xhtml" xml:lang="zh">
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<title>erf, erff, erfl</title>
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<body class="mediawiki ltr sitedir-ltr ns-0 ns-subject page-c_numeric_math_erf skin-cppreference2 action-view cpp-navbar">
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<div id="cpp-content-base">
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<div id="content"><a id="top"></a>
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<h1 id="firstHeading" class="firstHeading">erf, erff, erfl</h1>
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<div id="bodyContent">
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<div id="contentSub"><span class="subpages">< <a href="c.html">c</a>‎ | <a href="c-numeric.html">numeric</a>‎ | <a href="c-numeric-math.html">math</a></span></div>
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<div id="mw-content-text" lang="zh" dir="ltr" class="mw-content-ltr" xml:lang="zh">
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<table class="t-dcl-begin">
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<div>定义于头文件 <code><math.h></code></div>
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</td>
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<td></td>
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<td></td>
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<tr class="t-dcl t-since-c99">
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<div><span class="mw-geshi c source-c"><span class="kw4">float</span> erff<span class="br0">(</span> <span class="kw4">float</span> arg <span class="br0">)</span><span class="sy4">;</span></span></div>
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</td>
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<td>(1)</td>
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<td><span class="t-mark-rev t-since-c99">(C99 起)</span></td>
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<tr class="t-dcl t-since-c99">
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<td>
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<div><span class="mw-geshi c source-c"><span class="kw4">double</span> erf<span class="br0">(</span> <span class="kw4">double</span> arg <span class="br0">)</span><span class="sy4">;</span></span></div>
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</td>
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<td>(2)</td>
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<td><span class="t-mark-rev t-since-c99">(C99 起)</span></td>
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<tr class="t-dcl t-since-c99">
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<td>
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<div><span class="mw-geshi c source-c"><span class="kw4">long</span> <span class="kw4">double</span> erfl<span class="br0">(</span> <span class="kw4">long</span> <span class="kw4">double</span> arg <span class="br0">)</span><span class="sy4">;</span></span></div>
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</td>
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<td>(3)</td>
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<td><span class="t-mark-rev t-since-c99">(C99 起)</span></td>
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</tr>
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<tr class="t-dsc-header">
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<td>
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<div>定义于头文件 <code><tgmath.h></code></div>
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</td>
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<td></td>
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<td></td>
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<tr class="t-dcl t-since-c99">
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<div><span class="mw-geshi c source-c"><span class="co2">#define erf( arg )</span></span></div>
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</td>
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<td>(4)</td>
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<td><span class="t-mark-rev t-since-c99">(C99 起)</span></td>
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</tr>
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<tr class="t-dcl-sep">
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<td></td>
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<td></td>
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<td></td>
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</tbody>
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</table>
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<div class="t-li1"><span class="t-li">1-3)</span> 计算 <code>arg</code> 的<a href="https://en.wikipedia.org/wiki/Error_function" class="extiw" title="enwiki:Error function">误差函数</a>。</div>
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<div class="t-li1"><span class="t-li">4)</span> 泛型宏：若 <code>arg</code> 拥有 <span class="t-c"><span class="mw-geshi c source-c"><span class="kw4">long</span> <span class="kw4">double</span></span></span> 类型，则调用 <code>erfl</code> 。否则若 <code>arg</code> 拥有整数类型或 <span class="t-c"><span class="mw-geshi c source-c"><span class="kw4">double</span></span></span> 类型，则调用 <code>erf</code> 。否则调用 <code>erff</code> 。</div>
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<h3><span class="mw-headline" id=".E5.8F.82.E6.95.B0">参数</span></h3>
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<table class="t-par-begin">
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<tr class="t-par">
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<td>arg</td>
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<td>-</td>
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<td>浮点值</td>
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</tr>
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</table>
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<h3><span class="mw-headline" id=".E8.BF.94.E5.9B.9E.E5.80.BC">返回值</span></h3>
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若不出现错误，则返回 <code>arg</code> 的误差函数的值，即 <span class="mjax" style="display:none">\(\frac{2}{\sqrt{\pi} }\int_{0}^{arg}{e^{-{t^2} }\mathsf{d}t}\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<table>
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<tr>
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<td>2</td>
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</tr>
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<tr>
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<td><span class="t-mrad"><span>√</span><span>π</span></span></td>
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</tr>
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</table>
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<span style="font-size: 1.3em; vertical-align:-0.08em; margin-right: 0.08em;">∫</span><span class="t-su t-su-b">arg<br />
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0</span><i>e</i><sup>-t<span class="t-su">2<br /></span></sup>d<i>t</i> 。 若因下溢出现值域错误，则返回（舍入后的）正确结果，即 <span class="mjax" style="display:none">\(\frac{2\cdot arg}{\sqrt{\pi} }\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<table>
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<tr>
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<td>2*arg</td>
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</tr>
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<tr>
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<td><span class="t-mrad"><span>√</span><span>π</span></span></td>
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</tr>
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</table>
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。
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<h3><span class="mw-headline" id=".E9.94.99.E8.AF.AF.E5.A4.84.E7.90.86">错误处理</span></h3>
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<p>报告 <a href="c-numeric-math-math_errhandling.html"><tt>math_errhandling</tt></a> 中指定的错误。</p>
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<p>若实现支持 IEEE 浮点算术（ IEC 60559 ），则</p>
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<ul>
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<li>若参数为 ±0 ，则返回 ±0</li>
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<li>若参数为 ±∞ ，则返回 ±1</li>
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<li>若参数为 NaN ，则返回 NaN</li>
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</ul>
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<h3><span class="mw-headline" id=".E6.B3.A8.E8.A7.A3">注解</span></h3>
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<p>若 <span class="t-c"><span class="mw-geshi c source-c"><span class="sy3">|</span>arg<span class="sy3">|</span> <span class="sy1"><</span> <a href="c-types-limits.html"><span class="kw375">DBL_MIN</span></a><span class="sy2">*</span><span class="br0">(</span><a href="c-numeric-math-sqrt.html"><span class="kw665">sqrt</span></a><span class="br0">(</span>π<span class="br0">)</span><span class="sy2">/</span><span class="nu0">2</span><span class="br0">)</span></span></span> 则保证下溢。</p>
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<span class="mjax" style="display:none">\({\operatorname{erf}(\frac{x}{\sigma \sqrt{2} })\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">erf(<span class="t-mfrac"></span></span>
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<table>
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<tr>
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<td>x</td>
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</tr>
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<tr>
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<td>σ<span class="t-mrad"><span>√</span><span>2</span></span></td>
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</tr>
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</table>
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) 是测量结果小于与平均数相差 <span class="mjax" style="display:none">\(x\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">x</span> 的值的概率，其误差服从标准差为 <span class="mjax" style="display:none">\(\sigma\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">σ</span> 的正态分布。
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<h3><span class="mw-headline" id=".E7.A4.BA.E4.BE.8B">示例</span></h3>
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<div class="t-example">
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<div class="t-example-live-link"></div>
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<div dir="ltr" class="mw-geshi" style="text-align: left;">
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<div class="c source-c">
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<pre class="de1"><span class="co2">#include <stdio.h></span>
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<span class="co2">#include <math.h></span>
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<span class="kw4">double</span> phi<span class="br0">(</span><span class="kw4">double</span> x1, <span class="kw4">double</span> x2<span class="br0">)</span>
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<span class="br0">{</span>
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<span class="kw1">return</span> <span class="br0">(</span>erf<span class="br0">(</span>x2<span class="sy2">/</span><a href="c-numeric-math-sqrt.html"><span class="kw665">sqrt</span></a><span class="br0">(</span><span class="nu0">2</span><span class="br0">)</span><span class="br0">)</span> <span class="sy2">-</span> erf<span class="br0">(</span>x1<span class="sy2">/</span><a href="c-numeric-math-sqrt.html"><span class="kw665">sqrt</span></a><span class="br0">(</span><span class="nu0">2</span><span class="br0">)</span><span class="br0">)</span><span class="br0">)</span><span class="sy2">/</span><span class="nu0">2</span><span class="sy4">;</span>
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<span class="br0">}</span>
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<span class="kw4">int</span> main<span class="br0">(</span><span class="kw4">void</span><span class="br0">)</span>
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<span class="br0">{</span>
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<a href="c-io-puts.html"><span class="kw835">puts</span></a><span class="br0">(</span><span class="st0">"normal variate probabilities:"</span><span class="br0">)</span><span class="sy4">;</span>
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<span class="kw1">for</span><span class="br0">(</span><span class="kw4">int</span> n<span class="sy1">=</span><span class="sy2">-</span><span class="nu0">4</span><span class="sy4">;</span> n<span class="sy1"><</span><span class="nu0">4</span><span class="sy4">;</span> <span class="sy2">++</span>n<span class="br0">)</span>
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<a href="c-io-fprintf.html"><span class="kw850">printf</span></a><span class="br0">(</span><span class="st0">"[%2d:%2d]: %5.2f%%<span class="es1">\n</span>"</span>, n, n<span class="sy2">+</span><span class="nu0">1</span>, <span class="nu0">100</span><span class="sy2">*</span>phi<span class="br0">(</span>n, n<span class="sy2">+</span><span class="nu0">1</span><span class="br0">)</span><span class="br0">)</span><span class="sy4">;</span>
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<a href="c-io-puts.html"><span class="kw835">puts</span></a><span class="br0">(</span><span class="st0">"special values:"</span><span class="br0">)</span><span class="sy4">;</span>
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<a href="c-io-fprintf.html"><span class="kw850">printf</span></a><span class="br0">(</span><span class="st0">"erf(-0) = %f<span class="es1">\n</span>"</span>, erf<span class="br0">(</span><span class="sy2">-</span><span class="nu16">0.0</span><span class="br0">)</span><span class="br0">)</span><span class="sy4">;</span>
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<a href="c-io-fprintf.html"><span class="kw850">printf</span></a><span class="br0">(</span><span class="st0">"erf(Inf) = %f<span class="es1">\n</span>"</span>, erf<span class="br0">(</span>INFINITY<span class="br0">)</span><span class="br0">)</span><span class="sy4">;</span>
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<span class="br0">}</span></pre></div>
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</div>
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<p>输出：</p>
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<div dir="ltr" class="mw-geshi" style="text-align: left;">
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<div class="text source-text">
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<pre class="de1">normal variate probabilities:
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[-4:-3]: 0.13%
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[-3:-2]: 2.14%
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[-2:-1]: 13.59%
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[-1: 0]: 34.13%
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[ 0: 1]: 34.13%
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[ 1: 2]: 13.59%
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[ 2: 3]: 2.14%
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[ 3: 4]: 0.13%
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special values:
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erf(-0) = -0.000000
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erf(Inf) = 1.000000</pre></div>
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</div>
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</div>
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<h3><span class="mw-headline" id=".E5.BC.95.E7.94.A8">引用</span></h3>
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<div class="t-ref-std-11">
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<ul>
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<li>C11 标准（ISO/IEC 9899:2011）：</li>
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</ul>
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<dl>
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<dd>
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<ul>
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<li>7.12.8.1 The erf functions (p: 249)</li>
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</ul>
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</dd>
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</dl>
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<dl>
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<dd>
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<ul>
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<li>7.25 Type-generic math <tgmath.h> (p: 373-375)</li>
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</ul>
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</dd>
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</dl>
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<dl>
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<dd>
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<ul>
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<li>F.10.5.1 The erf functions (p: 525)</li>
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</ul>
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</dd>
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</dl>
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<div class="t-ref-std-c99">
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<ul>
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<li>C99 标准（ISO/IEC 9899:1999）：</li>
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</ul>
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<dl>
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<dd>
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<ul>
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<li>7.12.8.1 The erf functions (p: 230)</li>
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</ul>
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</dd>
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</dl>
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<dl>
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<dd>
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<ul>
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<li>7.22 Type-generic math <tgmath.h> (p: 335-337)</li>
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</ul>
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</dd>
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</dl>
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<dl>
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<dd>
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<ul>
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<li>F.9.5.1 The erf functions (p: 462)</li>
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</ul>
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</dd>
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</dl>
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</div>
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<h3><span class="mw-headline" id=".E5.8F.82.E9.98.85">参阅</span></h3>
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<table class="t-dsc-begin">
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="c-numeric-math-erfc.html"><span class="t-lines"><span>erfc</span><span>erfcf</span><span>erfcl</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-c99">(C99)</span></span><span><span class="t-mark-rev t-since-c99">(C99)</span></span><span><span class="t-mark-rev t-since-c99">(C99)</span></span></span></div>
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</div>
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</td>
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<td>计算补误差函数<br />
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<span class="t-mark">(函数)</span></td>
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</tr>
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</table>
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<h3><span class="mw-headline" id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5">外部链接</span></h3>
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<a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Erf.html">Weisstein, Eric W. "Erf."</a> From MathWorld--A Wolfram Web Resource.</div>
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</div>
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<div class="visualClear"></div>
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</div>
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</body>
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